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In this paper, we prove that for each $c\\geq 0$ and $b \\geq 2$, and for any $u$ in a cycle of $S_{[c,b]}$, (1) if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted int"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.0297","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-01T17:18:59Z","cross_cats_sorted":[],"title_canon_sha256":"0a8f4bb66b44e73cb21635cc5dca9b570d1af85b1d4a9f87ce1281efc47e59f5","abstract_canon_sha256":"078f9a8601bdf8f1a79bab2021aaac762cb94a3f595fdf23db091cb11a8bd35e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:41.820918Z","signature_b64":"evlyraUSRX3nQwS7cLHmeh3EvEQU9VYMp+UKoBid/JIJMKWXotM5Xpcpe3Qgzsl9lNMhdZaaXYaWErlif/shDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbc138bfba7340ecd72a06e0aefeafe9914e7a6b57628ed9c750e5b70401b5de","last_reissued_at":"2026-05-18T01:04:41.820535Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:41.820535Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Augmented generalized happy functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Breeanne Baker Swart, Christina Eubanks-Turner, Helen G. 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